The Ur-Dragon gets nervous when you start to mess with his tail.  Because he is often waving it around, it's sometimes difficult to target individual tail hearts.  I like to jump on the tail and slash away with 100/1,000 Kisses.  However, inevitably Ur-D will knock me off.  I don't believe it is possible to grapple the tail indefinitely.

When thrown or knocked off the tail, one usually lands behind the Ur-Dragon.  This sets you up for some excellent bow shots.  You can aim along the tail to center of mass.  You're essentially shooting at the side of the barn and are bound to hit something.  A Downpour/Hailstorm Volley can target the rear hearts.  Fivefold or Tenfold Flurry has spread decreasing the need for pinpoint accuracy.  Regular shots can be fired in rapid succession.  In other words, there are a number of techniques that will serve you well.  If you have Blast Arrows, all the better:

Shooting along the tail to Ur-D's Center of Mass00:45

Shooting along the tail to Ur-D's Center of Mass

If you want to quickly take out a tail heart, use the same technique at the beginning of a session and use a Maker's Finger.  If Ur-D twitches his tail as you are firing, you will still hit somthing of value and not waste a precious Maker's Finger.  

In the below video, the Ur-Dragon turns to face the party at the onset of the encounter.  Leila spots a tail heart that she decides to target.  The Ur-D becomes preoccupied with the pawns allowing Leila to slip behind him.  After shooting the tail heart, she goes up to it to verify that it was destroyed.  The place where the heart was is sill glowing a bit but the red glow fades quickly.  Slashing it yields only brown blood verifying that the heart is no more:

When Ur-D turns his back on you, give him the finger01:01

When Ur-D turns his back on you, give him the finger

The techique is the same regardless of the arrows being used.  Obviously, one wants to use more precision in selecting the target with the Maker's Finger.  For the purposes of the video, we wanted to verify that the tail heart was destroyed.  Under normal circumstances, we would take advantage of the Ur-D's collapse to select the best mounting point and exploit the success of the Maker's Finger.